Integration is the basic operation in
integral calculus. While
differentiation has straightforward
rules by which the derivative of a complicated
function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common
antiderivatives.
These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of
Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all
closed-form expressions have closed-form antiderivatives; this study forms the subject of
differential Galois theory, which was initially developed by
Joseph Liouville in the 1830s and 1840s, leading to
Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is e−x2, whose antiderivative is (up to constants) the
error function.
Since 1968 there is the
Risch algorithm for determining indefinite integrals that can be expressed in term of
elementary functions, typically using a
computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the
Meijer G-function.
Lists of integrals
More detail may be found on the following pages for the lists of
integrals:
Other useful resources include
Abramowitz and Stegun and the
Bateman Manuscript Project. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
There are several web sites which have tables of integrals and integrals on demand.
Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration.
Wolfram Research also operates another online service, the Mathematica Online Integrator.
Integrals of simple functions
C is used for an
arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of
antiderivatives.
These formulas only state in another form the assertions in the
table of derivatives.
Integrals with a singularity
When there is a
singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the
Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in
there is a singularity at 0 and the
antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[2]
Products of functions proportional to their second derivatives
Absolute-value functions
Let f be a
continuous function, that has at most one
zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then
where sgn(x) is the
sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a
piecewise constant function):
when n is odd, and .
when for some integer n.
when for some integer n.
when for some integer n.
when for some integer n.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every
interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen
step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:
There are some functions whose antiderivatives cannot be expressed in
closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008,
ISBN1-58488-956-X / 9781584889564.
Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002.
ISBN1-58488-291-3. (Many earlier editions as well.)
Integration is the basic operation in
integral calculus. While
differentiation has straightforward
rules by which the derivative of a complicated
function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common
antiderivatives.
These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of
Gradshteyn and Ryzhik. In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all
closed-form expressions have closed-form antiderivatives; this study forms the subject of
differential Galois theory, which was initially developed by
Joseph Liouville in the 1830s and 1840s, leading to
Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is e−x2, whose antiderivative is (up to constants) the
error function.
Since 1968 there is the
Risch algorithm for determining indefinite integrals that can be expressed in term of
elementary functions, typically using a
computer algebra system. Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the
Meijer G-function.
Lists of integrals
More detail may be found on the following pages for the lists of
integrals:
Other useful resources include
Abramowitz and Stegun and the
Bateman Manuscript Project. Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
There are several web sites which have tables of integrals and integrals on demand.
Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration.
Wolfram Research also operates another online service, the Mathematica Online Integrator.
Integrals of simple functions
C is used for an
arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of
antiderivatives.
These formulas only state in another form the assertions in the
table of derivatives.
Integrals with a singularity
When there is a
singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the
Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in
there is a singularity at 0 and the
antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −iπ when using a path above the origin and iπ for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[2]
Products of functions proportional to their second derivatives
Absolute-value functions
Let f be a
continuous function, that has at most one
zero. If f has a zero, let g be the unique antiderivative of f that is zero at the root of f; otherwise, let g be any antiderivative of f. Then
where sgn(x) is the
sign function, which takes the values −1, 0, 1 when x is respectively negative, zero or positive.
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g is here for insuring the continuity of the integral.
This gives the following formulas (where a ≠ 0), which are valid over any interval where f is continuous (over larger intervals, the constant C must be replaced by a
piecewise constant function):
when n is odd, and .
when for some integer n.
when for some integer n.
when for some integer n.
when for some integer n.
If the function f does not have any continuous antiderivative which takes the value zero at the zeros of f (this is the case for the sine and the cosine functions), then sgn(f(x)) ∫ f(x) dx is an antiderivative of f on every
interval on which f is not zero, but may be discontinuous at the points where f(x) = 0. For having a continuous antiderivative, one has thus to add a well chosen
step function. If we also use the fact that the absolute values of sine and cosine are periodic with period π, then we get:
There are some functions whose antiderivatives cannot be expressed in
closed form. However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008,
ISBN1-58488-956-X / 9781584889564.
Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press, 2002.
ISBN1-58488-291-3. (Many earlier editions as well.)